This material is adapted from the article “Microrobots
and Micromechanical Systems” by W. S. N. Trimmer, Sensors and Actuators,
Volume 19, Number 3, September 1989, pages 267 - 287, and other
sources. The book "Micromechanics and MEMS"
has this and other interesting articles on small mechanical systems; published
by the IEEE Press, number PC4390, ISBN 0-7803-1085-3. A more detailed
analysis of the scaling of electromagnetic forces is given in the Appendix
to “Microrobots and Micromechanical Systems.”

A nice description of scaling is given in “Trimmer’s Vertical
Bracket Notation” in the book “Fundamentals
of Microfabrication” by Marc Madou, ISBN 0-8493-9451-1, CRC Press 1997.

To design micromechanical actuators, it is helpful to
understand how forces scale. A simple notation for understanding
multiple force laws and equations is described below. This notation
is used to describe how different forces scale into the small (and large)
domain.

This paper uses a matrix formalism to describe the scaling
laws. This nomenclature shows a number of different force laws in a single
equation. In this notation, the size of the system is represented by a
single scale variable, S, which represents the linear scale of the system.
The choice of S for a system is a bit arbitrary. The S could be the separation
between the plates of a capacitor, or it could be the length of one edge
of the capacitor. Once chosen, however, it is assumed that all dimensions
of the system are equally scaled down in size as S is decreased (isometric
scaling). For example, nominally S = 1; if S is then changed to 0.1, all
the dimensions of the system are decreased by a factor of ten. A number
of different cases are shown in one equation. For example,

shows four cases for the force law. The top force law
scales as S, next scales as S squared or S2 (I hope this
appears as S superscript 2 on your screen, one never knows with the web),
the next as S3, and the bottom as S4.
The scaling of the time required to move an object using these forces is
given as

The top element in equation 2 is S1.5. This
is how the time scales when the force scales as S1. The
second element shows that t scales as S1 when the
force scales as S2. The third and forth element show how
the time scales when the force scales as S3 and S4
respectively. This notation is used consistently throughout this
paper. A dash [—] means that this case does not apply.

This vertical bracket notation can be used for other
scaling laws. For example, if one had a desire the top element could
refer to the case where the force scales as S6. Or the
top element could represent to the case where the acceleration scales as
S1, and the second element represent the case where the acceleration
scales as S2 , ... . These vertical brackets can be defined
for the convenience of the problem at hand. All that is needed is
the initial definition of what each element represents. (Equation
1 is this definition in our present case.)

Magnetic ForcesThis Section examines the scaling of magnetic forces
caused by the interactions of electrical currents. Three cases are examined:
A) constant temperature rise from the center to the exterior of the
coil windings, B) constant heat flow per unit surface
area of the coil windings, and C) constant electrical current density
in the coil windings. Assumption A) leads to forces that scale as
S2, assumption B) leads to forces that scale as S3,
and assumption C) leads to forces that scale as S4. These
three cases are depicted in equation 2.5. The derivation of this
force scaling requires a bit of math and will not be given here.
This derivation is given in the appendix of “Microrobots and Micromechanical
Systems.”

The above force scaling is for the case of two electrical
currents interacting. As S decreases, these forces decrease because
it is difficult to generate large magnetic fields with small coils of wire
(electromagnets). However permanent magnets maintain their strength
as they are scaled down in size, and it is often advantageous to design
magnetic systems that use the interaction between an electromagnet and
a permanent magnet. In the discussion below the scaling between two
electromagnets will be given in square brackets [ Sn ], and
the scaling between a permanent magnet and an electromagnet will be given
in curly brackets { Sn }.

Case C) Here the current density J is assumed to
be constant or J = S0, and hence a wire with one tenth the area
carries one tenth the current. The heat generated per volume of windings
is constant for this case. The force generated for this constant current
case scales as [ S4 ] { S3 }, i e., when the system
decreases by a factor of ten in size, the force generated by two interacting
electromagnets decreases by (1 / 10)4, or a factor of ten thousand.
Clearly this is not a strong micro force. (However, on the galactic
scale, magnetic forces become truly impressive. Looking at the spiral
arms of the S and SB galaxies, I wonder how these large magnetic fields
effect the complex twisting of galactic matter.)

Case B) Since heat can be more easily conducted
out of a small volume, it is possible to run isolated small motors with
higher current densities than assumed above. However, increasing the current
density makes the motors much less efficient. (Note, electronics
is usually much more wasteful of power than the micromechanical components,
and the power used by the motor is often insignificant.) If the heat flow
per unit surface area of the windings is constant, the current density
in the wires scales as J = S-0.5 This increase
in current density for small systems increases the force generated, and
the force scales as [ S3 ] { S2.5 } .

Case A) A third possible constraint on the magnetic
system as it is scaled down is the maximum temperature the wire and
insulation can withstand. If the system parameters are scaled so that there
is a constant temperature difference between the windings and surrounding
environment, then the current density scales as J = ( S-1 )
and the force scales as [ S2 ] { S2 } As will
be discussed later, forces that scale as ( S2 ) are useful in
small systems. Hence in many micro designs, it may be advantageous
to use the aggressive increase in current density assumed in case A.

In summary, the currents required for the different force
laws scale as:

These current scaling are the result of the assumptions
in Case A, Case B, and Case C) and generate the forces:

In designing micro electromagnets, one must also consider
electromigration. At high current densities, tiny wires are deformed
by the current and the wire can break. For example thin aluminum
interconnects at current densities higher than 5 x 105 A/ cm2
show the development of voids and hillocks which can lead to coil failure.
The temperature, composition and length of time the conductor is used have
a large effect on the electromigration. (References: [1] A.Scorzoni
et al., "Non-Linear Resistance Behavior in the Early Stages and After Electromigration
in Al-Si lines", J. Appl. Phys., 80 (1), p.143 (1996). and [2] A.Scorzoni,
I.De Munari and H. Stulens, "Non-Destructive Electrical Techniques as Means
for Understanding the Basic Mechanisms of Electromigration", MRS Symposia
Proceedings, Vol.337, pp.515--526 (1994).)

Electrostatic forcesElectrostatic actuators have a distinguished history,
but are not in general use for motors. (If you can, get a copy of
the delightful book by O. D. Jefimenko, "Electrostatic Motors," Published
by Electret Science Company, Star city, 1973. It is difficult to
find, but contains beautiful illustrations of early electrostatic motors.)
Electrostatic forces, however, become significant in the micro domain and
have numerous potential applications. The exact form of the scaling of
electrostatic forces depends upon how the E field changes with size. Generally,
the breakdown E field of insulators increases as the system becomes smaller.
Two cases will be examined here: (1) constant E field ( E= S0
); and (2) an E field that increases slightly as the system becomes smaller
(E = S-0.5 ). This second case exemplifies
the increasing E fields one can use as the system is scaled down.
(An early paper by Paschen discusses the increase in the breakdown
E field as a gap becomes smaller. F. Paschen, Uber die zum Funkenubergang
in Luft, Wasserstoff and Kohlensaure bei verschiedenen Drucken erforderliche
Potentialdifferenz. Annalen der Physick, 37:69-96, 1889. Also
Marc Madou's book "Fundamentals of Microfabrication" has a description
and plot of Paschen's curve on page 59.)For the constant electric field ( E = S0 )
the force scales as S2 When E scales as S-0.5
, then the force has the even better scaling of F = S1
. When the size of the system is decreased, both of these force laws
give increasing accelerations and smaller transit times.

Other forces

There are several other interesting forces. Biological
forces from muscle are proportional to the cross section of the muscle,
and scale as S2 Pneumatic and hydraulic forces are
caused by pressures (P) and also scale as S2.
Large forces can be generated in the micro domain using pressure related
forces. Surface tension has an absolutely delightful scaling of
S1 , because it depends upon the length of the interface.

The unit cube

Below is a discussion of how the above force laws affect
the acceleration, transit time, power generation and power dissipation
as one scales to smaller domains. In going from here to there as quickly
as possible with a certain force, one wants to accelerate for half the
distance, and then decelerate. The mass of the object scales as S3
(density is assumed to be intensive, or to not change with scale). Now
the acceleration is given by equations of dynamics as:

and the transit time is:

where SF represents the scaling
of the force F. Here only the time to accelerate has been calculated, but
an equal time is needed to decelerate, and both these times scale in the
same way. For the forces given in equation (1), the accelerations and transit
times can be expressed as

and

Even in the worst case, where F = S4
(the bottom element), the time required to perform a task remains
constant, t = S0 , when the system is scaled down. Under more
favorable force scaling, for example, the F = S2
scaling case, the time required decreases as t = S1 with
the scale. A system ten times smaller can perform an operation ten times
faster. This is an observation that we know intuitively: small things tend
to be quick.

Inertial forces tend to become insignificant in the small
domain, and in many cases kinematics may replace dynamics. This will probably
lead to interesting new control strategies.

Power generated and dissipatedAs the scale of a system is changed, one wants to know
how the power produced depends upon the force laws. For example, consider
the unit cube above, which is first accelerated and then decelerated. The
power, P, or the work done on the object per unit time is

The scaling of each of the terms on the right is known.

The power that can be produced per unit volume ( V=
S3 ) is

When the force scales as S2 then
the power per unit volume scales as S-1 . For example,
when the scale decreases by a factor of ten, the power that can be generated
per unit volume increases by a factor of ten. For force laws with a higher
power than S2 , the power generated per volume degrades
as the scale size decreases. There are several attractive force laws that
behave as S2, and one should try to use these forces
when designing small systems. (Please remember, these force laws depend
upon general assumptions, there is always room to be clever.)

For the magnetic case, one may be concerned about the
power dissipated by the resistive loss of the wires. The power due to this
resistive loss, PR, is

where A is the cross section of the wire, (rho) is the
resistivity of the wire, and L is the length of the wire. This gives

where (A L) is the volume. The resistivity scales as
S0 and the volume scales as S3
and from equation (3) above,

Hence the power dissipated scales as:

and the power per unit volume is:

For the magnetic case A) where force scales as S2,
the power that must be dissipated per unit volume scales as S-2
, or, when the scale is decreased by a factor of ten, a hundred times as
much power must be dissipated within a set volume. This magnetic case is
bad if one is concerned about power density or the amount of cooling needed.
If power dissipation or cooling are not a critical concern, then this scaling
case produces more substantial forces. In the future, superconductors may
give us stronger micro electromagnets.

Summary of the scaling results

The force has been found to scale in one of four different
ways: [ S1 ] , [ S2 ] , [ S3 ] , and [
S4 ] . If the scale size is decreased by a factor of ten,
the forces for these different laws decrease by ten, one hundred, one thousand,
and ten thousand respectively. In most cases, one wants to work with force
laws that behave as [ S1 ] or [ S2 ] .
The different cases that lead to these force laws, the accelerations, the
transit times, and the power generated per unit volume are given below.

and

For the force laws that behave as [ S1 ] or
[ S2 ] , the acceleration increases as one scales down the system.
The power that can be produced per unit volume also increases for these
two laws. The surface tension scales advantageously, [ S1 ]
, however, it is not clear how to use this force in most applications.
Biological forces also scale well, [ S2 ] but may be difficult
to implement. Electrostatic and pressure related forces appear to be quite
useful forces in the small domain.